Finite Group Databasebeta

← #16931757

Subgroups — #16931757

Order 4352.

726 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14533942C211yesmaximal, characteristic
2#14531137C211yesmaximal, characteristic
3#14527305C212yesmaximal, complemented
4#14526669C212yesmaximal, complemented
5C17×#1462C211yesmaximal, characteristic
6C2×#10587610C2211yescharacteristic
7C22×C22⋊Dic17—21no—
8#11762734C412yescomplemented
9#11762721C412yescomplemented
10#11762703—22no—
11#11762672—21no—
12#11762665—21no—
13C22⋊C4×Dic17—24no—
14C4×C22⋊Dic17—21no—
15C4×C22⋊Dic17C2211yescharacteristic
16C4×C22⋊Dic17C2212yes—
17#11762547—24no—
18C22×C22⋊C4×C17C2211yescharacteristic
19C17×#407C2212yes—
20C17×#323—22no—
21C25×C17D411yescharacteristic
22C23×C4×C17D411yescharacteristic
23C23×C4×C17C2311yescharacteristic
24C23×Dic17—22no—
25C22×Dic34C2×C411yescharacteristic, complemented
26C2×M16×C17—22no—
27C2×C22⋊C4×C17—22no—
28C2×C22⋊C4×C17—22no—
29C2×C22⋊C4×C17C2×C412yes—
30C2×C22⋊C4×C17D412yes—
31C2×C22⋊Dic17—28no—
32C2×C22⋊Dic17—22no—
33C2×C22⋊Dic17—22no—
34C2×C22⋊Dic17—22no—
35C2×C22⋊Dic17—21no—
36C2×C22⋊Dic17—21no—
37C2×C22⋊Dic17—22no—
38C2×C4⋊Dic17C2×C411yescharacteristic, complemented
39#10587610—22no—
40#10587610—22no—
41C2×#63225—21no—
42C2×C4×Dic17—22no—
43C2×C4×Dic17—21no—
44C2×C4×Dic17—22no—
45C17×C22⋊C8—22no—
46C17×C22⋊C8—22no—
47#10587502—22no—
48#10587502—22no—
49#10587502—22no—
50#10587502C2×C412yescomplemented
51#10587502D414yes—
52C24×C17—21no—
53C24×C17—24no—
54C24×C17—21no—
55C24×C17—42no—
56C24×C17—81no—
57C24×C17C22⋊C412yes—
58C24×C17C2×D411yescharacteristic
59C22×C4×C17—21no—
60C22×C4×C17—21no—
61C22×C4×C17—42no—
62C22×C4×C17—81no—
63C22×C4×C17C22⋊C412yes—
64C22×C4×C17C22×C411yescharacteristic
65C22×C4×C17C2×D412yes—
66C22×C4×C17C2×D411yescharacteristic
67C22×C4×C17#4112yes—
68C22×Dic17—24no—
69C22×Dic17—28no—
70C22×Dic17—21no—
71C22×Dic17—22no—
72C22×Dic17—21no—
73C22×Dic17—82no—
74C22×Dic17—82no—
75C2×Dic34—22no—
76C2×Dic34—81no—
77M16×C17—82no—
78C2×C8×C17—42no—
79C22⋊C4×C17—24no—
80C22⋊C4×C17—24no—
81C22⋊Dic17—28no—
82C22⋊Dic17—28no—
83C22⋊Dic17—42no—
84C22⋊Dic17—44no—
85C22⋊Dic17—41no—
86C22⋊Dic17—41no—
87C22⋊Dic17—82no—
88C4⋊Dic17—22no—
89#63225—42no—
90C4×Dic17—82no—
91C4×Dic17—82no—
92C4×Dic17—81no—
93#9407—171nomaximal, Sylow
94C23×C17—22no—
95C23×C17—22no—
96C23×C17—24no—
97C23×C17—22no—
98C23×C17—22no—
99C23×C17—22no—
100C23×C17—22no—
101C23×C17—41no—
102C23×C17—42no—
103C23×C17—42no—
104C23×C17—42no—
105C23×C17—42no—
106C23×C17—81no—
107C23×C17—81no—
108C23×C17—88no—
109C23×C17#10414yes—
110C23×C17C2×C22⋊C411yescharacteristic
111C23×C17#12011yescharacteristic
112C23×C17#12311yescharacteristic
113C2×C4×C17—22no—
114C2×C4×C17—42no—
115C2×C4×C17—41no—
116C2×C4×C17—42no—
117C2×C4×C17—41no—
118C2×C4×C17—42no—
119C2×C4×C17—42no—
120C2×C4×C17—81no—
121C2×C4×C17C4×D412yes—
122C2×C4×C17#12112yes—
123C2×Dic17—28no—
124C2×Dic17—44no—
125C2×Dic17—44no—
126C2×Dic17—42no—
127C2×Dic17—42no—
128C2×Dic17—42no—
129C2×Dic17—41no—
130C2×Dic17—41no—
131C2×Dic17—42no—
132C2×Dic17—42no—
133C2×Dic17—88no—
134Dic34—82no—
135C8×C17—82no—
136#2561—171no—
137#2273—171no—
138#1508—172no—
139#1462D1711yescharacteristic
140#1297—172no—
141C22×C17—21no—
142C22×C17—21no—
143C22×C17—21no—
144C22×C17—21no—
145C22×C17—21no—
146C22×C17—24no—
147C22×C17—21no—
148C22×C17—21no—
149C22×C17—44no—
150C22×C17—44no—
151C22×C17—44no—
152C22×C17—44no—
153C22×C17—42no—
154C22×C17—42no—
155C22×C17—82no—
156C22×C17—84no—
157C22×C17#38611yescharacteristic
158C22×C17C2×#10412yes—
159C22×C17#45512yes—
160C4×C17—22no—
161C4×C17—22no—
162C4×C17—42no—
163Dic17—48no—
164Dic17—82no—
165Dic17—81no—
166Dic17—81no—
167C2×C22⋊Q8—171no—
168C22×C22⋊C4—341no—
169C22×C22⋊C4C2×D1711yescharacteristic
170#407C2×D1712yes—
171#397—172no—
172#393—172no—
173#388—342no—
174#381—341no—
175#379—341no—
176C4×C22⋊C4—171no—
177C4×C22⋊C4—172no—
178C4×C22⋊C4—344no—
179C4×C22⋊C4—341no—
180#328—344no—
181#323—22no—
182C2×C17—22no—
183C2×C17—22no—
184C2×C17—41no—
185C2×C17—44no—
186C2×C17#188511yescharacteristic
187C2×C17#189611yescharacteristic
188C2×C17#191311yescharacteristic
189C25D6811yescharacteristic
190C22×Q8—171no—
191C23×C4—342no—
192C23×C4D6811yescharacteristic
193C23×C4C22×D1711yescharacteristic
194C2×M16—22no—
195C22⋊Q8—342no—
196C22⋊Q8—342no—
197C2×C4⋊C4—171no—
198C2×C4⋊C4—341no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—342no—
202C2×C22⋊C4—348no—
203C2×C22⋊C4—342no—
204C2×C22⋊C4—341no—
205C2×C22⋊C4—341no—
206C2×C22⋊C4—342no—
207C2×C22⋊C4—342no—
208C2×C22⋊C4C4×D1712yes—
209C2×C22⋊C4C22⋊D1712yes—
210C2×C42—342no—
211C2×C42—342no—
212C2×C42—341no—
213C22⋊C8—22no—
214C22⋊C8—22no—
215#95—172no—
216#95—174no—
217#95—342no—
218#95—342no—
219#95—342no—
220C17#940711yescharacteristic, complemented, Sylow
221C24—21no—
222C24—24no—
223C24—21no—
224C24—42no—
225C24—81no—
226C24#6322712yes—
227C24C2×D6811yescharacteristic
228C2×Q8—342no—
229C2×Q8—1361no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—42no—
233C22×C4—81no—
234C22×C4—344no—
235C22×C4—348no—
236C22×C4—342no—
237C22×C4—341no—
238C22×C4—341no—
239C22×C4—1362no—
240C22×C4—1362no—
241C22×C4#6322712yes—
242C22×C4C2×C4×D1711yescharacteristic
243C22×C4D4×D1712yes—
244C22×C4#6325412yes—
245C22×C4C2×C22⋊D1711yescharacteristic
246M16—82no—
247C2×C8—42no—
248C4⋊C4—342no—
249C4⋊C4—682no—
250C22⋊C4—24no—
251C22⋊C4—24no—
252C22⋊C4—348no—
253C22⋊C4—348no—
254C22⋊C4—682no—
255C22⋊C4—684no—
256C22⋊C4—681no—
257C22⋊C4—681no—
258C22⋊C4—1362no—
259C42—1362no—
260C42—1361no—
261C42—1362no—
262C23—22no—
263C23—22no—
264C23—24no—
265C23—22no—
266C23—22no—
267C23—22no—
268C23—22no—
269C23—41no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—81no—
275C23—81no—
276C23—88no—
277C23#1058749614yes—
278C23#1058756811yescharacteristic
279C23#1058757211yescharacteristic
280C23C2×#6322711yescharacteristic
281Q8—1362no—
282C2×C4—22no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—81no—
290C2×C4—348no—
291C2×C4—684no—
292C2×C4—684no—
293C2×C4—682no—
294C2×C4—682no—
295C2×C4—682no—
296C2×C4—682no—
297C2×C4—682no—
298C2×C4—681no—
299C2×C4—681no—
300C2×C4—1368no—
301C2×C4#1058756712yes—
302C2×C4#1058762512yes—
303C8—82no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—24no—
311C22—21no—
312C22—44no—
313C22—44no—
314C22—44no—
315C22—44no—
316C22—42no—
317C22—42no—
318C22—82no—
319C22—84no—
320C22#1176268611yescharacteristic
321C22C2×#1058749612yes—
322C22#1176298912yes—
323C4—22no—
324C4—22no—
325C4—42no—
326C4—688no—
327C4—1362no—
328C4—1361no—
329C4—1361no—
330C2—22no—
331C2—22no—
332C2—41no—
333C2—44no—
334C2#1452917011yescharacteristic
335C2#1452922511yescharacteristic
336C2#1452932411yescharacteristic
subgroupG/H
centerC22#11762686
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#1462C2
socleC22×C17#386
prime radicalC25×C17D4
cosocleC23×C4×C17C23
hypercenter#1462D17
hypocenterC17#9407
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1462

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #14527305C2semidirect
4. #14526669C2semidirect
8. #11762734C4semidirect
9. #11762721C4semidirect
25. C22×Dic34C2×C4semidirect
38. C2×C4⋊Dic17C2×C4semidirect
50. #10587502C2×C4semidirect
220. C17#9407semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#145339422
#145311372
#145273052
#145266692
C17×#14622
#940717